Carbon is a nonmetal, and graphite, which is an allotrope of carbon, conducts electricity. Why?

In order for a substance to conduct electricity, it must contain moving, charged particles such as electrons. 

Usually, when a carbon atom is part of a molecule, all four of its valence electrons are occupied in covalent bonds with other atoms. The electrons in covalent bonds are shared between atoms. This means that none of carbon's valence electrons are free to move around the molecule. Therefore, these types of molecules do not conduct electricity.

Graphite is composed of sheets of carbon atoms. The carbon sheets are held together by weak dispersion forces. The carbon atoms within the sheets are covalently bonded together; however, only three of carbon's valence electrons are involved in covalent bonds with other carbon atoms. The fourth valence electron in each carbon atom is free to become delocalized and travel over the surface of the carbon sheets. Since graphite contains moving charged particles, it is able to conduct electricity.